I am trying to prove the following theorem :
A discontinuous, fixed point free group G of isometries of $\mathbb{R}^2$ is generated by one or two elements.
We may find an outline of the demonstration here although it only consider the case of the group containing only translations.
If $G$ contains only translations the theorem follows from the demonstration found in the link above. When G contains both glide reflections and translations Stillwell's hint is :
If $G$ is a discontinuous,fixed point free group of glide reflections and translations,let g be a glide reflection of minimal lenght in $G$ and let h be an element of minimal lenght not in the direction of g. Show that g, h must have perpendicular directions (e.g. by finding shorter elements when the directions of g ,h are not perpendicular.
As in the previous case I tried to show that when g and h are not in perpendicular direction then I can find a $m\mathbb{Z}; |h^{-1}g^{m}(P)-P|<|h(P)-p|$ but having little information about this composition I got little success in this approach. Any hint is appreciated.
Definition 1: A fixed point free group of isometries is a group $G$ such that for all $g\in G-{e}$ and $x\in \mathbb{R}^{2}$ we have $g(P)\neq P$. Here $e$ is the identity.
Definition 2: A glide reflection is the composition of a reflection in a line and a translation along that line.
Let $L: Isom(E^2)\to O(2)$ be the homomorphism given by taking linear parts of the affine maps. Suppose that $f, g\in Isom(E^2)$ are glide-reflections along non-perpendicular (and non-parallel lines). Then their linear parts $L(f), L(g)$ are reflections in non-perpendicular and distinct lines through the origin. It is a pleasant exercise to show that such $L(f), L(g)$ do not commute. Therefore, $$ L([f,g])= [L(f), L(g)]\ne id. $$ Hence, $[f,g]$ is a rotation.